Math 1180
Summary of topics and practice problems for Midterm 3
1. Relationshp between two random variables
- Compute joint distributions from marginal and conditional distributions (7.1)
- Compute mrginal and conditional distributions from a joint distrbution (7.1)
- check joint distribution for independence
- Compute covariance (7.2)
- Compute correlation (7.2)
- Interpret near-zero, near-one, near-negative-one correlation (7.2)
2. Expected value, variance and covariance of random variable combinations (7.3)
- Expected value of a sum
- expected value of a product
- varince of a sum
- expected value, variance and covariance of variables multiplies by a
constant
3. Binomial distribution (7.5)
4. Geometric distribution (7.6)
5. POisson distribution (7.7)
6. Exponential distribution (7.6)
For each distribution you need to be able to: - Recognize the distribution in a variety of settings
- Computing the correpsonding probabilities using the formulas
- Finding the mean and variance
7. Normal distribution (7.8,7.9)
- Computing probabilities from Normal distribution by reducing to the
Standard Normal probabilities and using the table
- Marking probabilities on the graph
- Find distributions for sums and averages of random variables using
Central Limit Theorem
8. Approximations
- Use POisson distribution to compute binomial probabilities
- Use normal distribution to compute poisson and binomial probabilities
1. A staining technique has a 20% chance of success. Let Sn be the ranom
variable representing the number of successful staining in n attempts.
a) What are the conditions for Sn to have a binomial distribution?
b) Under the binomial distribution conditions, find expected value and
standard deviation of S10
c) Find P r(S10 = 2), using the fact that 10
= 45
2
2. Each of two guitarists only knows A and D chords. On each beat each
guitarist plays an A chord with probability 0.75, and a D chord with probablity 0.25. These two probabilities are assumed the same in all subsequent
questions.
a) If they begin by playing independently, give the joint distribution
b)If the second player plays an A with probability 90% when the second
player does, give the joint distribution
c) For the following joint distribution, find marginal distributions.
P r(AA) = 0.1, P r(AD) = 0.5, P r(DA) = 0.3, P r(DD) = 0.1
d) If playing A is worth 1 point and playing D is worth 0, find the covariance,
and correlation in case of distribution from c)
3. A study finds that the ice cream sales and the rate of drowning deaths
have a correlation of .85.
a) Is this a strong correlation?
b) Based on this information what would you predict happened with ice
cream sales last month if we know that the rate of drowning deaths increased
sharply?
c) Can we conclude that ice cream consumption causes drowning?
4. N1 is the number of successful shooting attempts by player F. He
usually hits the target with probability 0.8. N2 is the number of successful
attempts by player G who hits each target with probability 0.4.
a) If player F has 10 attempts, and player G has 3 attempts, what is
TOTAL expected number of successful hits from both players?
b) If player F gets 3 points for each of the hits, what is the expected
number and variance of number of points he will earn?
c) If the player F still gets 3 points, and the player G gets 2 points for
each time he scores, what is the expected number of TOTAl points they will
get together? If the players shoot independently, what is the variance of the
TOTAl score? If the independency assumption is removed, can you compute
the variance?
5. On average, there are 222 sunny days per year in Salt Lake City.
Assuming that the weather on each day is independent of other days, and
that any day can be sunny with the same probability, find probability that
at least 2 days among the next 10 will be sunny. Today is rainy. How long
will I have to wait on average to the next rainy day?
6. If a student scores 10 on the exam with probability .1, scores 5 with
probability .7 and scores 0 with probability .2, find the probability that an
average grade in 100 student class is above 5, using central limit theorem.
7. Action potentials of a neuron can be thought of as a POisson process
with rate of 20 Hz. Find probability that the time between second and third
action potential is longer than 0.01 but shorter than 0.06 seconds. How many
spikes to we expect in a 0.5 sec interval? What is the probability that the
number of spikes in a 0.5 sec interval is less than expected?
8. A measurement result is normally distributed with mean 0.1 and standard eviation 0.05. How often will we get a negative measurement? (use the
table at the back of the book) What is the probability that the measurement
falls between 0.15 and 0.2? Mark associated probability on the graph of the
pdf.
Formulae list
P r(A|B) =
P r(A ∩ B)
P r(B)
P r(A ∩ B) = P r(A|B)P r(B)
Pr(A) = Σni=1 P r(A|Ei)P r(Ei)
P r(A|B)P r(B)
P r(A)
P r(B|A) =
X̄ = E(X) = Σni=1 xi p(xi )
X̄ = E(X) =
Z
b
a
xf (x)dx
MAD = Σni=1 |xi − X̄|p(xi )
MAD =
Z
b
|x − X̄|f (x)dx
a
V ar = σ 2 = Σni=1 x2i p(xi ) − X̄ 2
V ar = σ 2 =
Cov(X, Y ) =
Z
b
a
m X
n
X
j=1 i=1
x2 f (x)dx − X̄ 2
(xi − X̄)(yj − Ȳ )pij
Cov(X, Y ) = E(XY ) − X̄ Ȳ
ρX,Y =
Cov(X, Y )
σX σY
E(X + Y ) = E(X) + E(Y )
E(aX) = aE(X)
E(XY ) = E(X)E(Y ) + Cov(X, Y )
V ar(X + Y ) = V ar(x) + V ar(Y ) + 2Cov(X, Y )
V ar(aX) = a2 V ar(X)
Cov(aX, Y ) = Cov(x, aY ) = a · Cov(X, Y )
!
n k
p (1 − p)n−k
P r(N = k) = b(k; n, p) =
k
E(N) = np, V ar(N) = np(1 − p)
P r(T = t) = gt = p(1 − p)t−1
P r(T ≤ t) = Gt = 1 − (1 − p)t
1
1−p
E(T ) = , V ar(t) =
p
p2
(λt)k e−λt
k!
E(N) = λt, V ar(N) = λt
P r(N = k) = p(k; λ, t) =
T : f (t) = λe−λt ; F (t) = 1 − e−λt
E(T ) =
1
1
, V ar(T ) = 2
λ
λ
Sn ∼ N(nµ, nσ 2 )
An ∼ N(µ, σ 2 )
X −µ
σ
(x−µ)2
1
e− 2σ2
f (x) = √
2πσ 2
Z=
A copy of table from the book